# Sharedchao

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Example Calculations

*.sharedChao

Example calculations below will be performed using data from the Eckburg 70.stool_compare files with an OTU definition of 0.03.

Estimating the richness of shared OTUs between two communities. A Non-parametric richness estimator of the number of shared OTUs between two communities has been developed that is analogous to the Chao1 (6) single community richness estimator. The $S_{A,B Chao} \mbox{(11)$ estimator is calculated as:

$S_{A,B Chao} = S_{12 \left ( Obs \right )} + f_{11} \frac {f_{1+}f_{+1}}{4f_{2+}f_{+2}} + \frac{f_{1+}^{2}}{2f_{2+}} + \frac{f_{+1}^{2}}{2f_{+2}}$,

where,

$f_{11}$ = number of shared OTUs with one observed individual in A and B

$f_{1+}, f_{2+}$ = number of shared OTUs with one or two individuals observed in A

$f_{+1}, f_{+2}$ = number of shared OTUs with one or two individuals observed in B

$f_{\left(rare \right)1+}$ = number of OTUs with one individual found in A and less than or equal to 10 in B.

$f_{\left(rare \right)+1}$ = number of OTUs with one individual found in B and less than or equal to 10 in A.

$n_{rare}$ = number of sequences from A that contain less than 10 sequences.

$m_{rare}$ = number of sequences from B that contain less than 10 sequences.

$S_{12\left(rare\right)}$ = number of shared OTUs where both of the communities are represented by less than or equal to 10 sequences.

$S_{12\left(abund\right)}$ = number of shared OTUs where at least one of the communities is represented by more than 10 sequences.

$S_{12\left(obs\right)}$ = number of shared OTUs in A and B.

Calculation of the $S_{A,B Chao}$ requires the number of OTUs where only one sequence was observed from each library, $f_{11}$. For our example case, $f_{11}$ is 2. Plugging the f-values and $S_{12\left( obs \right )}\mbox{into } S_{A,B Chao}$ yields a value of 76.7, which matches the value in $S_{A,B Chao}$ of the table above.